Differential Equations Summary
1. First order differential equations
a. Variables Separable DE:
Arrange through manipulation such that the form below is achieved:
dyygdxxf )()(
=
Integrate subsequently to yield the required solution.
Example: Solve
y
dx
dy −= 1
for y<1.
SOLUTIO :
1
1
1
1=
−
⇒−= dx
dy
y
y
dx
dy
∫ ∫
=
−
−
−dxdy
y1
1
Cxy
+
=
−
−
|1|ln
Since
,1
<
y
(
)
Cxy +=−− 1ln
Bx
ey
+−
=−1
∴
x
Aey
−
−= 1 where
B
eA
=
,
cB
−
=
(shown)
This solution is commonly termed the GEERAL SOLUTIO, where
A
is
unknown. When initial conditions are provided, eg
y
=0 when
x
=0, then
A
assumes a specific value and the solution is termed the PARTICULAR SOLUTIO.
When we use the GC to plot out a series of graphs for various values of
A
, the result
is that we produce a family of solution curves.
b. Reduction through substitution:
The introduction of an intermediate variable aids in reducing the original
differential equation to a far simpler version which is readily solvable.
Example: Use the substitution y=
vx
, where
v
is a function of x, to solve the
differential equation yx
dx
dy
x
+=
3
.
SOLUTIO :
dx
dv
xv
dx
dy
vxy
+=⇒=
Substituting into the differential equation gives
vxx
dx
dv
xvx +=
+3
